Special Program Learning Seminar

Two flows are said to be Kakutani equivalent if one is isomorphic to the other after time change, or equivalently if there are Poincare sections for the flows so that the respective induced maps are isomorphic to each other. Ratner showed that if $G=SL(2,R)$ and $\Gamma$ is a lattice in $G$, and if $u_t$ is a one parameter unipotent subgroup in $G$ then the $u_t$ action on $G/\Gamma$ equipped with Haar measure is loosely Bernoulli, i.e. Kakutani equivalent to a circle rotation. Thus any two such systems are Kakutani equivalent to each other. On the other hand, Ratner showed that if $G=SL(2,R)\times SL(2,R)$ and $\Gamma$ is a reducible lattice, and $u_t$ is the diagonally embedded one parameter unipotent subgroup in $G$, then $(G/\Gamma, u_t, m)$ is not loosely Bernoulli.

This leads naturally to the question: which unipotent flows are Kakutani equivalent to one another? How much of the structure is preserved after a time change?

Joint work with Daren Wei.